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VOOZH | about |
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VOOZH | about |
Sequences, series, and summations are fundamental concepts of mathematical analysis, and they have practical applications in science, engineering, and finance.
It is a set of numbers in a definite order according to some definite rule (or rules). Each number of the set is called a term of the sequence, and its length is the number of terms in it. We can write the sequence as: . A finite sequence is generally described by a1, a2, a3.... an, and an infinite sequence is described by a1, a2, a3.... to infinity. A sequence {an} has the limit L and we write:
or as
For example:
If the terms of a sequence can be described by a formula, then the sequence is called a progression.
1, 1, 2, 3, 5, 8, 13, ....., is a progression called the Fibonacci sequence in which each term is the sum of the previous two numbers.
Theorem 1: Given the sequence if we have a function f(x)such that f(n) = and then . This theorem tells us that we take the limits of sequences much like we take the limit of functions.
Theorem 2: (Squeeze Theorem): If for all n > N for some N and then
Theorem 3: If then .
Note that for this theorem to hold the limit must be zero and it won’t work for a sequence whose limit is not zero.
Theorem 4: If and the function f is continuous at L, then
Theorem 5: The sequence is convergent if and divergent for all other values of r. Also, This theorem is a useful theorem giving the convergence/divergence and value (for when it’s convergent) of a sequence that arises on occasion.
If and are convergent sequences, the following properties hold:
provided
A series is simply the sum of the various terms of a sequence. If the sequence is a1, a2, a3, ... , an the expression a1 + a2 + a3 + ... + an is called the series associated with it. A series is represented by 'S' or the Greek symbol . The series can be finite or infinite. Examples:
If the sequence of partial sums is a convergent sequence (i.e. its limit exists and is finite) then the series is also called convergenti.e. if then . Likewise, if the sequence of partial sums is a divergent sequence ( if
or its limit doesn’t exist or is plus or minus infinity) then the series is also called divergent.
If and be convergent series then
If and be convergent series then
If be convergent series then
If and be convergent series then if for all n N then
Theorem 1 (Comparison test): Suppose for for some k. Then
(1) The convergence of implies the convergence of
(2) The convergence of implies the convergence of
Theorem 2 (Limit Comparison test): Let and , and suppose that . Then converges if and only if converges.
Theorem 3 (Ratio test): Suppose that the following limit exists, . Then,
(1) If converges
(2) If diverges
(3) If might either converge or diverge
Theorem 4 (Root test): Suppose that the following limit exists:, . Then,
(1) If converges
(2) If diverges
(3) If might either converge or diverge
Theorem 5 (Absolute Convergence test): A series is said to be absolutely convergent if the series converges.
Theorem 6 (Conditional Convergence test): A series is said to be conditionally convergent if the series diverges but the series converges .
Theorem 7 (Alternating Series test): If , and , the 'alternating series' will converge.
Summation is the addition of a sequence of numbers. It is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable. The summation symbol, , instructs us to sum the elements of a sequence. A typical element of the sequence which is being summed appears to the right of the summation sign.
where c is any number. So, we can factor constants out of a summation.
So we can break up a summation across a sum or difference.
Note that while we can break up sums and differences as mentioned above, we can’t do the same thing for products and quotients. In other words,
, for any natural number .
. If the argument of the summation is a constant, then the sum is the limit range value times the constant.
Various examples of summation formula includes:
